Questions from General Calculus

Q: Use the formula from Exercise 41(a) to prove

Use the formula from Exercise 41(a) to prove Exercise 41(a): (a). Suppose f is a one-to-one differentiable function and its inverse function f-1 is also differentiable. Use implicit differentiation...

Q: (a). Sketch the graph of the function f (x

(a). Sketch the graph of the function f (x) = sin (sin-1 x). (b). Sketch the graph of the function g (x) = sin-1 (sin x), x ∈ R (c). Show that g'(x) = cos x/|cos x|. (d). Sketch the graph of h (x) = c...

Q: Show that the ellipse x2/a2+ y2/b2 =

Show that the ellipse x2/a2+ y2/b2 = 1 and the hyperbola x2/A2 – y2/B2 = 1 are orthogonal trajectories if A2 < a2 and a2 – b2 = A2 + B2 (so the ellipse and hyperbola have the same foci).

Q: Find the value of the number a such that the families of

Find the value of the number a such that the families of curves y = (x + c)-1 and y = a (x + k)1/3 are orthogonal trajectories.

Q: (a). The van der Waals equation for n moles of

(a). The van der Waals equation for n moles of a gas is where P is the pressure V, is the volume, and T is the temperature of the gas. The constant R is the universal gas constant and a and b are posi...

Q: A semicircle with diameter PQ sits on an isosceles triangle PQR to

A semicircle with diameter PQ sits on an isosceles triangle PQR to form a region shaped like a two-dimensional ice-cream cone, as shown in the figure. If A (θ) is the area of the semicirc...

Q: Find the exact value of each expression. tan (sin

Find the exact value of each expression. tan (sin-1 (2/3)

Q: Find the exact value of each expression. csc (arccsc3

Find the exact value of each expression. csc (arccsc3/5)

Q: (a). Evaluate limx→∞ x sin 1/x.

(a). Evaluate limx→∞ x sin 1/x. (b). Evaluate limx→0 x sin 1/x. (c). Illustrate parts (a) and (b) by graphing y = x sin (1/x).

Q: Find the exact value of each expression. sin (2

Find the exact value of each expression. sin (2 tan-1 √2)